1. Field of the Invention
The present invention relates to a model and accompanying algorithm to simulate and analyze ink ejection from a piezoelectric print head. More particularly, the model of this invention includes a quadrilateral grid for finite-difference-based ink-jet simulation where the algorithm is designed to solve a set of partial differential equations for two-phase flows that have been newly developed on the quadrilateral grid. The quadrilateral grid is transformed to a uniform square grid, and the derivatives of various parameters (e.g., the velocities, pressure, and level set) in the newly developed partial differential equations are calculated on the uniform square grid. A stable and powerful numerical algorithm is developed to solve the derived equations. The simulation model may be embodied in software, hardware or combination thereof and may be implemented on a computer or other processor-controlled device.
2. Description of the Related Art
Results of computational fluid dynamics (CFD) ink-jet simulation have been very useful in the design of piezoelectric ink-jet print heads. FIG. 1 shows how a practical inkjet simulation may be carried out. An analytical tool such as an equivalent circuit 11 receives as an input the dynamic voltage to be applied to the piezoelectric PZT actuator and simulates the ink behavior under the influence of the ink cartridge, supply channel, vibration plate, and PZT actuator. That is, from the input voltage and an ink flow rate, the equivalent circuit 11 calculates an inflow pressure that drives the CFD code 12. The CFD code 12 then solves the governing partial differential equations, i.e., the incompressible Navier-Stokes equations for two-phase flows, for fluid velocity, pressure and interface position, and feeds back the ink flow rate to the equivalent circuit. The sequence is repeated as long as needed.
A CFD code has been used by Seiko Epson to solve the Navier-Stokes equations. Such code employs the volume of fluid method (VOF) to take into account the ink-air interface. VOF performs fairly well with regard to mass conversation but is not so accurate on surface tension aspects of fluid flow, especially when the ink droplet is smaller than 5 pico liters. However, since the capability of ejecting ultra small ink droplets is essential for any photo quality ink-jet printer today, an improved modeling method which included the level set method was adopted by Seiko Epson to more accurately capture the ink-air interface in CFD simulations. Since there is a mathematical relation between the level set and the interface curvature, and hence the surface tension, the level set method excels whenever surface tension is important.
Because solving the level set equation by finite element analysis usually results in a serious mass conservation problem, finite difference analysis is usually the best choice among numerical schemes to be used with the level set method. FIG. 2 shows a typical rectangular grid for a finite difference analysis. Since the wall of the narrowing section of the modeled nozzle is not parallel to any coordinate axis, the discretized computational domain can not faithfully fit the real nozzle wall. The body-fitted quadrilateral grid in FIG. 3 does not have that problem. While a non-rectangular quadrilateral grid like the one in FIG. 3 can be naturally handled in finite element analysis, performing a finite difference analysis on such a grid is very difficult.
The first finite difference scheme for solving the incompressible Navier-Stokes equations on quadrilateral grids works well for single-phase fluids on a two-dimensional Cartesian coordinate system. The scheme was later extended to an axisymmetric coordinate system. It was not clear, however, how to extend the algorithm to two-phase flows, to include surface tension, and to include the level set convection equation.